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Introduction

Any closed, simple, piecewise tex2html_wrap_inline1406 curve tex2html_wrap_inline1408 defines a Hamiltonian flow, the billiard flow inside the billiard table T [29, 32, 31, 16]. The billiard map of the table T is the Poincaré map tex2html_wrap_inline1414 defined by the standard cross-section of the billiard flow. The study of the billiard map for a convex table T goes back to Birkhoff [2] and the theory is still far from being completed. In this work we assume that T is convex and tex2html_wrap_inline1406 and use the standard facts about the phase space tex2html_wrap_inline1422 of the billiard map tex2html_wrap_inline1424 .

tex2html_wrap1666 tex2html_wrap1668
A homotopically nontrivial tex2html_wrap_inline1406 invariant circle tex2html_wrap_inline1428 defines a closed curve tex2html_wrap_inline1372 in the interior of T. It is called the caustic. While in general the caustics of T may be a complicated curve, the caustics in this paper are convex. The invariant circle tex2html_wrap_inline1436 consists then of the supporting rays of tex2html_wrap_inline1372 and the table T is obtained from tex2html_wrap_inline1372 by the string construction [29, 31]: One takes an unstretchable string having length tex2html_wrap_inline1378 , wraps it around tex2html_wrap_inline1372 , pulls it tight at a point M and drags it around tex2html_wrap_inline1372 . The point M then traces the boundary tex2html_wrap_inline1376 of the table.

tex2html_wrap1670 tex2html_wrap1672
The geometric description of an ellipse with given foci is a special case of the string construction.

Let now tex2html_wrap_inline1460 be a closed convex curve with perimeter tex2html_wrap_inline1462 . By varying the string length tex2html_wrap_inline1464 we obtain all billiard tables tex2html_wrap_inline1376 with the caustic tex2html_wrap_inline1372 . The corresponding billiard maps tex2html_wrap_inline1470 form a one-parameter family of twist maps which is determined by tex2html_wrap_inline1372 . Our investigatigation of this family emphasis on the relationship between the geometric shape of tex2html_wrap_inline1372 and the dynamics of the family tex2html_wrap_inline1374 of billiard maps.

Let tex2html_wrap_inline1478 be the invariant circle of tex2html_wrap_inline1374 corresponding to tex2html_wrap_inline1372 . Due to the time-reversal symmetry, there are two invariant circles in tex2html_wrap_inline1422 associated with tex2html_wrap_inline1372 . We make a choice of tex2html_wrap_inline1384 by requiring that the rotation number of tex2html_wrap_inline1384 is less than or equal to 1/2. In the special case when tex2html_wrap_inline1494 is an interval, the tables tex2html_wrap_inline1496 are the ellipses with foci tex2html_wrap_inline1498 and tex2html_wrap_inline1500 . The invariant circle tex2html_wrap_inline1384 is then the lower of the two invariant circles in tex2html_wrap_inline1422 with the rotation number 1/2 (see Figure 3).

tex2html_wrap1674 tex2html_wrap1676
Let tex2html_wrap_inline1510 be the restriction of the twist map tex2html_wrap_inline1374 to tex2html_wrap_inline1384 . With a uniform parameter on tex2html_wrap_inline1384 , the maps tex2html_wrap_inline1518 form a continuous monotone family of orientation preserving circle homeomorphisms. The rotation number tex2html_wrap_inline1386 is a continuous, nondecreasing function which contains significant amount of information on the family tex2html_wrap_inline1374 of twist maps. For instance, tex2html_wrap_inline1524 is identically 1/2 if and only if tex2html_wrap_inline1372 is an interval or a point. In all other cases, one has tex2html_wrap_inline1530 and tex2html_wrap_inline1532 . If tex2html_wrap_inline1372 is a polygon with n sites then tex2html_wrap_inline1538 . Otherwise tex2html_wrap_inline1540 .

The invariant circle tex2html_wrap_inline1478 moves monotonically upward in the phase space as tex2html_wrap_inline1378 runs from tex2html_wrap_inline1462 to infinity (Figure 4).

tex2html_wrap1678 tex2html_wrap1680
For small tex2html_wrap_inline1378 , the curve tex2html_wrap_inline1384 is near the bottom tex2html_wrap_inline1560 of tex2html_wrap_inline1422 . As tex2html_wrap_inline1378 increases, tex2html_wrap_inline1384 moves up and approaches for tex2html_wrap_inline1568 the equator tex2html_wrap_inline1570 . It remains in general below the equator but not necessarily strictly as the ellipse shows.

We state now the main results of our work. There are two theorems. Our first result is that there is a Baire generic set tex2html_wrap_inline1572 of caustics tex2html_wrap_inline1372 for which the rotation function tex2html_wrap_inline1576 is a devil's staircase. More precisely, tex2html_wrap_inline1578 has the property that for any rational p/q, the interval tex2html_wrap_inline1582 has a nonempty interior. Some geometrical properties of tex2html_wrap_inline1372 imply that tex2html_wrap_inline1586 has nonempty interior. We apply this to show that tex2html_wrap_inline1572 contains all polygons and curves with a flat point.

In the second theorem, we study then the motion of Birkhoff periodic orbits of a fixed rotation number in the phase space tex2html_wrap_inline1422 . Let tex2html_wrap_inline1586 be the interval of the set of parameters tex2html_wrap_inline1378 for which there exists a Birkhoff periodic orbit on the curve tex2html_wrap_inline1384 . For tex2html_wrap_inline1598 , all Birkhoff periodic orbits with rotation number p/q are above tex2html_wrap_inline1384 . For tex2html_wrap_inline1604 , they are below tex2html_wrap_inline1384 . Global index arguments imply that for tex2html_wrap_inline1608 the curve tex2html_wrap_inline1384 separates the set of Birkhoff periodic orbits into two nonempty subsets. We prove that for tex2html_wrap_inline1598 near tex2html_wrap_inline1614 a Birkhoff periodic set of index -1 approaches tex2html_wrap_inline1384 from above. This set continues as a set of index +1 below tex2html_wrap_inline1384 for tex2html_wrap_inline1624 near tex2html_wrap_inline1614 . At tex2html_wrap_inline1628 , a bifurcation occurs and two Birkhoff periodic sets of index -1 are created on tex2html_wrap_inline1384 . The picture for tex2html_wrap_inline1378 near tex2html_wrap_inline1636 is obtained by an obvious symmetry from the case at tex2html_wrap_inline1614 : a Birkhoff periodic set of index +1 approaches from above the curve tex2html_wrap_inline1384 , collides with two Birkhoff periodic sets on tex2html_wrap_inline1384 and leaves tex2html_wrap_inline1384 as a Birkhoff periodic set of index -1. (see Figures 12,13,14).

The theme of relations between the shape of the billiard table and the billiard dynamics is well reflected in the literature. Here, we mention only a few papers which are especially close to our direction.

A problem which is closely related to our work is how to decide whether a specific billiard table has an invariant circle or not. We mention the theorem of Mather [19] about the global absence of invariant circles in the case when the billiard table has a flat point, the result of Hubacher [13] about the absence of invariant circles near the table if the curvature of the table has a discontinuity. An other related result is the theorem of Gruber [6, 7] which tells that generically in the Hausdorff topology on the set of convex tables, a billiard table has no caustics. On the other hand, if the table is smooth enough and the curvature is strictly positive, Lazutkin's result [17] assures in this case the existence of infinitely many caustics near the boundary of the table. The inverse problem of finding billiard tables with a special invariant circle or caustics is interesting both from the geometrical and from the dynamical point of view. Bialy [1] showed that if the phase space of the billiard map is foliated by a continuous family of homotopically non-trivial invariant circles, then the billiard table is a disc. There are billiard tables with flat invariant circles different from the equator [8]. The size of the area which is free of caustics can be estimated from the shape of the table [9]. For the billiard in an ellipse, there are caustics for any rotation number and the table can be obtained from any of them by the string construction. Elliptical billiards are believed to be special. According to a conjecture attributed to Birkhoff and stated by Poritsky [26], they are the only integrable billiards in the sense that a subset of full Lebesgue measure of the phase space is foliated by invariant circles. This Birkhoff-Poritsky problem is still open.

The paper is organized as follows. In section 2, we establish notation and collect preliminary results. In section 3, we investigate the rotation function and show that it is Baire generically a devil's staircase. If there exists some tex2html_wrap_inline1378 for which every orbit on tex2html_wrap_inline1384 is periodic, a caustic tex2html_wrap_inline1372 is called exceptional. We will show in section 4 that this property is quite restrictive. While tables with exceptional caustics with rotation number 1/3 have been constructed in [14] and his construction probably generalizes to any possible rational number, it is not known whether a table exists which has two exceptional invariant caustics. In section 5, we investigate the passage of the Birkhoff periodic orbits through the invariant circle tex2html_wrap_inline1384 as tex2html_wrap_inline1378 varies from tex2html_wrap_inline1462 to tex2html_wrap_inline1664 .

Next: Preliminaries Up: Billiards that share a Previous: Billiards that share a

Oliver Knill
Wed Jul 8 11:57:32 CDT 1998